Theory of (2+1)-dimensional fermionic topological orders and fermionic/bosonic topological orders with symmetries

We propose a systematic framework to classify (2+1)-dimensional (2+1D) fermionic topological orders without symmetry and 2+1D fermionic/bosonic topological orders with symmetry $G$. The key is to use the so-called symmetric fusion category $\mathcal{E}$ to describe the symmetry. Here, $\mathcal{E}=\mathrm{sRep}\left({\mathbb{Z}}_2^f\right)$ describing particles in a fermionic product state without symmetry, or $\mathcal{E}=\mathrm{sRep}\left(G^f\right) [\mathcal{E}=\mathrm{Rep}(G\left)\right]$ describing particles in a fermionic (bosonic) product state with symmetry $G$. Then, topological orders with symmetry $\mathcal{E}$ are classified by nondegenerate unitary braided fusion categories over $\mathcal{E}$, plus their modular extensions and total chiral central charges. This allows us to obtain a list that contains all 2+1D fermionic topological orders without symmetry. For example, we find that, up to $p+\mathrm{i}p$ fermionic topological orders, there are only four fermionic topological orders with one nontrivial topological excitation: (1) the $K=\left(\begin{array}{cc}\hfill -1& \hfill 0\\ \hfill 0& \hfill 2\end{array}\right)$ fractional quantum Hall state, (2) a Fibonacci bosonic topological order stacking with a fermionic product state, (3) the time-reversal conjugate of the previous one, and (4) a fermionic topological order with chiral central charge $c=\frac{1}{4}$, whose only topological excitation has non-Abelian statistics with spin $s=\frac{1}{4}$ and quantum dimension $d=1+\sqrt{2}$.